Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
When does the time for Udhiyah begin?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The time for Udhiyah (sacrificial offering) begins once the sun has risen on the day of Eid al-Adha—which is the tenth of Dhul-Hijjah—and a period of time has passed equivalent to two brief prayer units (Raka'at) and two brief sermons (Khutbah). It then continues until the sunset of the last of the days of Tashreeq, which are the eleventh, twelfth, and thirteenth of Dhul-Hijjah.
The Prophet (peace be upon him) said: "All the mountain passes of Mina are places of sacrifice, and in all the days of Tashreeq there is slaughtering." [Narrated by Al-Bayhaqi and Ibn Hibban].
The best time to slaughter is after finishing the Eid prayer, due to the saying of the Prophet (peace be upon him):
"Indeed, the first thing we begin with on this day of ours is to pray, then we return and slaughter. Whoever does that has attained our Sunnah, and whoever slaughters before [the prayer], it is only meat he has presented to his family; it is not part of the ritual sacrifice (Nusuk) in any way." [ٌReported by Bukhari & Muslim].
It is valid to sacrifice at any time, whether by night or by day; however, it is disliked (Makruh) at night. And Allah the Almighty knows best.
What are the Sunnahs and etiquettes recommended for the person offering the Udhiyah?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There are several Sunnahs and etiquettes that are recommended for the person offering the Udhiyah (sacrifice) to observe, including:
1-It is Sunnah for the one intending to sacrifice to refrain from removing any hair or nails once the first ten days of Dhul-Hijjah begin. The Prophet (peace be upon him) said: "When the ten days [of Dhul-Hijjah] begin and one of you intends to offer a sacrifice, let him not touch [remove] any of his hair or skin." [Narrated by Muslim]. If someone does remove any hair or nails, they have not committed a sin, and their sacrifice remains valid.
2-It is recommended for the person offering the sacrifice to slaughter the animal themselves. If they are unable to do so, they should witness the slaughter. The Messenger of Allah (peace be upon him) said to Fatimah (may Allah be pleased with her): "Stand and witness your sacrifice, for indeed, with its first drop of blood, your [previous sins] are forgiven." [Narrated by Al-Bayhaqi and Al-Tabarani; it is classified as a weak (da'if) hadith].
3-The animal should be positioned to face the Qiblah at the time of slaughter, as the Qiblah is the most noble of directions.
4-The slaughterer should say: "Bismillah ar-Rahman ar-Rahim" (In the name of Allah, the Entirely Merciful, the Especially Merciful). Even if one forgets to say it, the sacrifice is still permissible to eat. Allah (Blessed and Exalted be He) says: "So eat of that [meat] upon which the name of Allah has been mentioned" [Al-An'am: 118]. It is also recommended to send blessings upon the Messenger of Allah (peace be upon him) and to say "Allahu Akbar" (Allah is the Greatest) after the Tasmiyah.
5-The slaughterer should pray for the sacrifice to be accepted by saying: "O Allah, this is from You and for You, so please accept it from me." (Allahumma hadhihi minka wa ilayka, fataqabbal minni). And Allah the Almighty knows best.
Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.